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Constant Derivatives and the Power Rule

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Constant Derivatives and the Power Rule

The power rule is a fantastic "shortcut" for finding the derivatives of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort - often in your head!

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- Khan Academy: Calculus: Derivatives 3


In this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.

The Derivative of a Constant

Theorem: If f(x) = c where c is a constant, then f^{\prime}(x) = 0

Proof: f'(x)= \lim_{h \to 0}\frac {f(x+h)-f(x)}{h}=\lim_{h \to 0}\frac{c-c}{h} = 0

Theorem: If c is a constant and f is differentiable at all x , then \frac {d}{dx}[cf(x)] = c\frac {d}{dx}[f(x)] . In simpler notation (cf)^{\prime} = c(f)^{\prime} = cf^{\prime}

The Power Rule

Theorem: (The Power Rule) If n is a positive integer, then for all real values of x
\frac {d}{dx}[x^n] = nx^{n-1} .

Example A

Find f^{\prime} (x) for f(x)=16


If f(x) = 16 for all x , then f^{\prime} (x) = 0 for all x
We can also write \frac{d}{dx}16 = 0

Example B

Find the derivative of f(x)=4x^3


\frac {d}{dx}\left [{4x^3} \right] ..... Restate the function
4 \frac{d}{dx}\left [{x^3} \right] ..... Apply the Commutative Law
4 \left [{3x^2} \right] ..... Apply the Power Rule
12x^2 ..... Simplify

Example C

Find the derivative of f(x)=\frac{-2}{x^{4}}


\frac {d}{dx} \left [\frac{-2}{x^4} \right] ..... Restate
\frac {d}{dx}\left [{-2x^{-4}} \right] ..... Rules of exponents
-2 \frac {d}{dx}\left [{x^{-4}} \right] ..... By the Commutative law
-2 \left [{-4x^{-4-1}} \right] ..... Apply the Power Rule
-2 \left [{-4x^{-5}} \right] ..... Simplify
8x^{-5} ..... Simplify again
\frac {8}{x^5} ..... Use rules of exponents


A theorem is a statement accepted to be true based on a series of reasoned statements already accepted to be true. In the context of this lesson, a theorem is a rule that allows a quick calculation of the derivative of functions of different types.

A proof is a series of true statements leading to the acceptance of truth of a more complex statement.

Guided Practice


Find the derivatives of:

1) f(x)=x^{3}

2) f(x)=x

3) f(x)=\sqrt{x}

4) f(x)=\frac{1}{x^{3}}


1) By the power rule:

If f(x) = x^3 then f(x) = (3)x^{3-1} = 3x^2

2) Special application of the power rule:

\frac {d}{dx}[x] = 1x^{1-1} = x^0 = 1

3) Restate the function: \frac {d}{dx}[\sqrt{x}]

Using rules of exponents (from Algebra): \frac {d}{dx}[x^{1/2}]
Apply the Power Rule: \frac {1}{2}x^{1/2-1}
Simplify: \frac {1}{2}x^{-1/2}
Rules of exponents: \frac{1}{2x^{1/2}}
Simplify: \frac {1}{2\sqrt{x}}

4) Restate the function: \frac {d}{dx}\left [ \frac{1}{x^3} \right ]

Rules of exponents: \frac {d}{dx}\left [{x^{-3}} \right ]
Power Rule: -3x^{-3-1}
Simplify: -3x^{-4}
Rules of exponents: \frac {-3}{x^4}


  1. State the Power Rule.

Find the derivative:

  1. y = 5x^7
  2. y = -3x
  3. f(x) = \frac{1} {3} x + \frac{4} {3}
  4. y = x^4 - 2x^3 - 5\sqrt{x} + 10
  5. y = (5x^2 - 3)^2
  6. given y(x)= x^{-4\pi^2} when  x = 1
  7. y(x) = 5
  8. given u(x)= x^{-5\pi^3} what is  u'(2)
  9.  y = \frac{1}{5} when  x = 4
  10. given d(x)= x^{-0.37} what is  d'(1)
  11.  g(x) = x^{-3}
  12. u(x) = x^{0.096}
  13. k(x) = x-0.49
  14.  y = x^{-5\pi^3}

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